To define Besov spaces, we need the following dyadic unity partition
(see e.g. [1]}). Choose two nonnegative radial
functions \(\varrho \) , \(\varphi \in C^{\infty }(\mathbb {R}^{d})\) 
supported respectively in the ball \(\lbrace \xi \in \mathbb {R}^{d}:|\xi |\le \frac{3}{4} \rbrace \)  and the shell \(\lbrace \xi \in \mathbb {R}^{n}: \frac{3}{4}\le |\xi |\le \frac{8}{3} \rbrace \)  such that
\(\varrho (\xi )+\sum _{j\ge 0}\varphi (2^{-j}\xi )=1, \quad \forall \xi \in \mathbb {R}^{d}; \qquad \sum _{j\in \mathbb {Z}}\varphi (2^{-j}\xi )=1, \quad \forall \xi \ne 0.\) 
